Electrical analogues



Oct. 2, 1951 A WOLF 2,569,510

ELECTRICAL ANALOGUES Filed Dec. .15, 1947 v 2 Sheets-Sheet l PROBE 2?F76. ELECTRICAL C/RC ms 204 2a 22 2/ (VD/CATOR 4/ DECADE RES/S TANCL'80X 38 DECADE RES/S TANCE BOX 39 fill/FIE}? CIRCUIT 40 FIG. 3.

EQU/PO TENT/AL LINES INVENTOR. ALEXANDER WOLF A T TORNE V Oct. 2, 1951A. WOLF ELECTRICAL ANALOGUES 2 sheets-sheet 2 Filed Dec.

mm 3 63 qxs huzoq Q a s INVENTOR. AL EXANDER WOL F A T TORNE Y PatentedOct. 2, 1951 ELECTRICAL ANALOGUES Alexander Wolf, Houston, Tex.,assignor to The Texas Company, New York, N. Y., a corproation ofDelaware 1 Application December 15, 1947, Serial No. 791,797

1 Claim. (Cl. 346-139) This invention is concerned with the analysis ofmechanical and electrical systems which obey Laplaces equation. It findsa variety of applications, including production control from undergroundfluid reservoirs, such as oil and gas fields.

Ohms law for the flow of electrcity is expressed v by the equationmensional space taken at right angles to each other.

If Equations 1 and 3 are combined, the result is Laplaces equation, viz.

2 and and a This equation governs the flow of electricity 1: l.@ (1) ina homogeneous isotropic conducting medium,

5x 10 but in view of the analogy between the flow of a where I is thecurrent flowing through the unit fluid (which does not suiier change involume) of area of a section whose specific resistivity is in a porousmedia and the flow of electricity in p, and across which there is avoltage gradient the conductor, it may be taken as governing such 8Efluid flows as well, i. e. the flow of electricity is E is analogous tothe flow of fluids if the fluid is con- (1; is normal to the area).

Darceys law for. the flow of fluids through porous media is where V isthe rate of flow of a fluid whose viscosity is a through a unit area ofa section having a permeability k, and across which there is a sideredeither as incompressible or uncompressed.

There are a number of engineering problems involving the investigationof steady-state dynamic systems in which force distribution can beexpressed in terms of electrical potentials and in which Laplacesequation, as a practical approximation, may be considered ascontrolling. It has been proposed heretofore, to solve some of theseproblems (particularly that involving fluid flow in a porous medium) byconstructing an electrical pressure gradient analogue. Such an analoguemay take the form 6p of a conductive model, say a pool of electrolyte,

3; the shape of which is analogous to the system to be investigated,exterior forces operating upon g i gi to the tff l 1 hen the systembeing represented in the analogue by e w aws arel en a w electricalpotentials imposed across the model.

1 The conductor of the model may be of the elec- 5 tronic type, i. e. aconductive solid. Electrons are introduced at one or more points in themodel the reciprocal Of specific resistivity, is made nuand dis lacefree electrons throughout the conmericauy equal to the ratio ofpermeability ductor, so that electrons are forced to move out viscosity,in which case fluid flow becomes directat another point with resultantcurrent fl w 1y comparable to current flow and electrical po- Theconductor may also be of the ionic type, say tential becomes directlycomparable to pressure. a pool f electrolyte or an electrolyte dispersedElectricity, if it be likened to a moving fluid, 40 in a body of gel,current fl w being dependent is incompressible. Consequently Ohms law(Equation 1) may be combined with the equation of continuity for anuncompressed or incompressible fluid (i. e. one which does not sufferchange in volume). The equation can be expressed as follows, for theflow of electricity upon the mobility of ions through the conductor, butwith current flow and potential drop established in the conductor justas in the electronic conductors.

In both types of conductors, potential and potential gradients may bedetermined by means of probes in contact with the points in question andconnected to a potential measuring device such as a galvonometer.However, the collection of data by such means heretofore has beentedious and time consuming, with the result that electrical analoguesfor solution of problems of the type described have not been employed tothe fullest extent.

The exploitation of an oil or gas field may be accomplished either bythe expansion of the gas and oil in the producing formation, bydisplacement of the gas and oil by another fluid, or by simultaneousexpansion and displacement. Whenever the displacement of the oil or gasis the controlling factor there arises the problem of mapping theprogress of the boundary between the fluid in place and the displacingfluid. This problem is of particular interest in the operation of acycling project in a gas-condensate field. In such an operation wet gasmay be produced from one or more wells which are commonly calledextraction or output wells. The wet gas is sent to a processing plantwhere liquid condensate is removed. The dry gas remaining after removalof the liquid condensate is injected back into the producing formationthrough one or more wells, called injection or input wells, both toconserve the gas for future use and also to maintain the pressure in thefield. In such a cycling operation it is essential to anticipate themanner in which the dry gas will spread through the field, because theultimate recovery of wet gas, and therefore of valuable liquid products,depends largely on keeping the dry gas from breaking into the extractionwells until substantially complete production of wet gas has beenachieved. It is thus necessary to know the shape of the boundary orfluid interface between the wet and the dry gas around the injectionwells for various arrangements of wells and for various injection andextraction rates, so that a scheme can be selected which will postponethis break-through to the latest possible date while maintaining a givenrate of extraction and thereby permit the maximum recovery of wet gas.After break-through of dry gas into an extraction well has occurred, itthen may be necessary to determine the expected proportion of dry gaswhich will be mixed with the wet gas produced by the well.

Exploitation problems of the type described above and the application ofelectrical analogues to the solution thereof are discussed by Muskat inhis book The Flow of Homogeneous Fluids through Porous Media.Unfortunately the mathematical solution of the type of problem outlinedabove can be carried out only-for certain idealized arrangements ofinjection and extraction wells, and then only for the simplest ofboundary conditions. The mathematical vsolution for the wellarrangements and boundaries which are encountered in actual practice isentirely impractical because of the excessive labor involved. The onlyfeasible approach to the solution of these problems is through the useof models on a reduced scale. These models need not actually employ aporous medium and a fluid.

' As pointed out above, there exists, with certain assumptions, an exactanalogy between the flow of fluid in a porous medium and the flow ofelectrical current in a conducting body of similar geometry.

If electrical currents proportional to the rates of injection andextraction of fluid are passed through such a conductor by means ofelectrodes located at points corresponding to the positions of the wellsin thefield, then the electrical potential distribution in the conductoris exactly analogous to the fluid pressure distribution in the field,and the current flow lines correspond to the fluid flow lines in thefield. The study of fluid flow in an oil field is thus reduced to astudy of currentflow in an electrical conductor of suitable shape.

In making studies on electrical models employing electrolytic solutionsthe mapping of the fluid interface may be carried out by either of twodifferent known methods. The first of these methods, which has beenapplied quite widely, is by means of the ionic conduction or so-calledelectrolytic model. The electrolytic model consists essentially of anaqueous solution of an electrolyte held in a suitable porous medium toprevent excessive velocities on the part of the ions due to normaldiffusion. The electrolyte may be any ionizable salt. The porous mediummay be ordinary blotting paper or cardboard, or may be a hydrophilic gelmade from gelatin, agar-agar, or similar colloidal substances.

A color tracer is added to the electrolytic system to permit visualobservation of the progress of the ions between the points of differentelectrical potential in the system. The color tracer may be a hydrogenion indicator, such as phenolphthalein, contained in the electrolytesolution. in which case the negative electrodes represent the fluidsource or injection wells in the corresponding flow system and thepositive electrodes represent the extraction wells. The advance ofnegative ions as revealed by a color change of the indicator correspondsto the advancing fluid interface in the producing formation. In place ofan indicator a colored ion may be injected into the system at theelectrode corresponding to the injection well. In such a case acolorless solution of zinc-ammonium chloride may be employed as theelectrolyte in the porous medium and a solution of copper-ammoniumchloride may be injected at the positive electrodes. The progress of theblue copper-ammonium ions from the positive to the negative electrodescorresponds to the advancing fluid interface in the formation. Theprogress of the equivalent fluid interface may be observed and recordedby making photographs of the position of the colored areas of the modelat various time intervals after the experiments are begun.

The electrolytic model gives the desired results quite rapidly but withinferior accuracy due to a loss of sharpness of the boundaries of thefluid interface as the pattern spreads. It has been shown that the totaleffect of the several inherent errors of the electrolytic method mayresult in an overall error of as much as 25% in the determination of thevolume of the formation flooded by individual injection wells. In viewof this the results obtained with the electrolytic model are generallyconsidered to be of a qualitative nature or at best onlysemi-quantitative.

A second method for carrying out such studies is by means of theelectrical conduction or socalled potentiometric model. This modelconsists ordinarily of a pool of a conducting liquid. such as a dilutesolution of copper sulphate in water. The bottom of the conducting poolis shaped such that the depth of the conducting liquid at any pointcorresponds to the actual quantity of oil or gas contained in theproducing sand at the corresponding point in the formation. Theperiphery of the conducting pool is shaped to correspond to 'thegeometrical boundaries of the formation. The necessary information forthe construction of the conducting pool is obtained from an isopachousmap of the formation on which the contour lines of equal sand thicknesshave been converted to represent actual oil content by correcting forsand volume and connate water content. For practical reasons thereduction in scale is much larger in the horizontal than in the verticaldirection.

At points corresponding to the location of the various wells in thefield vertical metallic electrodes are introduced into the pool, usuallythrough the bottom so as not to interfere with the measurements. Throughthese electrodes electric currents are passed into and out of the pool,the flow of current being either into or out of the pool depending uponwhether the well represented is an injection or extraction well, and themagnitude of the currents being proportional to the fluid injection andextraction rates which are employed or which it is proposed to employ inthe exploitation of the field. As already mentioned, the direction ofcurrent flow at any point in the conducting pool is then identical tothe direction of fluid flow at the corresponding point in theoil-bearing formation, and the potential gradient at any point in theconducting pool is proportional to the pressure gradient at thecorresponding point in the formation. Any element of fluid in theformation follows a path corresponding to a current flow line in theconducting pool and the transit time for such element of fluid from onepoint to another in the formation is proportional to the line integraltaken along a flow line in the formation, where a: is the traveldistance and P is the pressure. Hence the transit time for any elementof fluid in the formation is also proportional to the line integraltaken along a current line in the conducting pool, where x is thecorresponding travel distance in the pool and V is the potential.

The problem of mapping the progress of an interface between the drivingand the driven fluids is essentially the problem of determining theseintegrals along all of the current lines in the electrically conductingpool. The potentiometric method, however, has not been employedextensively, because heretofore the means for obtaining the lineintegrals have been too cumbersome for practical application. The methodwhich has been employed previously for arriving at fluid travel time bymeans of potentlometric model studies is tedious and time consuming.

Application of my invention to oil field exploitation problems permitsthe collection of accurate data in but a small fraction of the timeheretofore involved, this being only one of a number of types ofproblems in which the invention offers advantages. Thus I have developeda mechanism which greatly facilitates the collection and utilization ofdata with electrical analogues of steady-state systems that aregoverned, at least approximately, by Laplace's equation. The invention,in essence, contemplates the combination which comprises a threedimensional electrically conductive model analogous to the system to beinvestigated, a chart (say one corresponding to a plan of the model),means for establishing across the model a current analogous to forceoperating upon the systom, a conductive probe disposed in contact withthe model, means for determining the potential of the probe, a markerdisposed adjacent the chart, and a linkage connecting the marker to theprobe for moving the marker on the chart to correspond to movements ofthe probe an the model.

The linkage may take only one of a number of forms, for example, an armconnecting the probe to the marker and supporting means which permit thearm to be moved both longitudinally and laterally, say in a planeparallel to that of the chart. It may also take the form of acombination of levers arranged as a pantograph connecting the probe tothe marker, so that movement of one produces a corresponding movement ofthe other.

The probe member may have but a single electrical contact, with acorresponding single tracing point on the marker. Preferably (asdisclosed in co-pending application Serial No. 674,904, filed June 6,1946, now abandoned) multiple contacts are provided on the probe, withcorresponding multiple tracing points on the marker, the linkage beingsuch that the tracing points move on the chart to correspond to themovements of the contacts on the model. Thus, for example, if it isdesired to plot lines of equipotential in the model, the probe may beequipped with a rotatable head upon which a pair of contacts arefastened in spaced relationship. Similarly the marker member is providedwith a rotatable head, provided with a pair of tracing points spaced tocorrespond to the contacts, the axes of rotation of the two headsbearing similar relationships respectively to the model and the chart.The mechanical linkage is such that the two heads move in unison andcorrespondingly respectively over model and chart and rotatecorrespondingly so that when the contacts are on an equipotential linein the model, with no potential difference indicated by a galvanometerconnected across them, a corresponding line will be indicated by thetracing points on the chart. The tracing points may be arranged to plotcurrent flow lines on the chart at right angles to lines ofequipotential discovered by the contacts of the probe.

A more comprehensive concept of my invention may be had from thefollowing detailed description, taken in conjunction with theaccompanying drawings in which:

Fig. 1 is a pictorial view or a simple form of the apparatus of theinvention equipped with a single probe or contact and a. single markeror tracing point with a pantograph linking the two;

Fig. 2 is a wiring diagram illustrating an electrical circuit for usewith the apparatus of Fig.

Fig. 3 is a diagram illustrating the relationship of equipotential linesto flow lines;

Fig. 4 is a diagram illustrating a modified form of linkage betweenprobe and marker; and

Fig. 5 is an isopachous map of the oil bearing formation of Fig. 4, withflow lines thereon constructed in accordance with the invention.

The apparatus of Figs. 1 and 2 employs a wooden tank 20 made up of aseries of plywood laminations. The interior or basin 2! of the tank isshaped to correspond to a wet gas field undergoing investigation, itsinterior being sealed by a waterproof insulating varnish coated with alayer of wax. Electrodes 22, 23, 24 representing injection wells andelectrodes 25, 26 representing extraction wells are 1 inch diametercopper gms. copper sulfate (anhydrous) 5 gms. sulfuric acid 5 gms. ethylalcohol 100 gms. distilled water The current at any electrode should bekept low enough to avoid formation of gas bubbles thereon with resultantincrease in contact resistance and shift of equipotential lines. It hasbeen found that currents of the order of .04 ampere per electrode can betolerated without appreciable efiect from gas bubbling with an'electrode depth of one inch. Larger diameter electrodes or greaterelectrode depth permit larger currents.

A probe 21 in the form of an electrical conductor is mounted on the endof a system of levers arranged as a one-to-one pantograph 28 having 1 afixed pivot 28A. The other end of the pantograph carries a tracer point29 disposed above an isopachous map 30 corresponding to the gas fieldrepresented by the electrolytic model 20. The map is so disposed thatwhen the probe occupies a given position on the electrolytic model themarker or tracer point occupies a corresponding position on the map.

The electrical details of the system are shown on Fig. 2. Current to theelectrodes is supplied by, a one-to-one isolation transformer 3| from a110 volt 60 cycle power line. The three electrodes 22, 2 3, 24representin injection wells are connected respectively through suitablevariable f resistors to one side of the transformer output winding'whilethe two electrodes 25, 2B representing extraction wells are connected tothe oppositeside of this transformer winding through similar sets ofresistors.

The resistances are adjusted in each case to give currents to theelectrodes corresponding to the chosen injection or extraction rate,measurement being made by means of a voltmeter 35,

which can be connected into any one of the series of resistances bymeans of single pole multiple throw switches 36, 31. l A potentiometerconsisting of two precision decade resistance boxes 38, 39 is' connected.between the injection electrode 24 and the extraction electrode 26. (Inpractice, it is best to select When the probe occupies a point on thisequipotential line, the input to the amplifier is a minimum as indicatedby a null on a galvanometer I connected in the output of the amplifier.

Each time that a. point on a particular equipotential line is located bythe probe in the electrolyte, the tracer point,-due to the action of thepantograph, occupies a corresponding point on the map. The tracer pointis then depressed to make that position and the operation is repeateduntil sufficient points on that equipotential line are marked. Thesetting of the decade boxes is then changed and another equipotentialline is similarly located, the points along each line being joined ineach case on the map.

Once the equipotential lines are constructed, a system of flow lineswhich are at all points normal tothe equipotential lines may beconstructed. This is illustrated in Fig. 3 wherein a flow line 42 hasbeen constructed normal to two equipotential lines 43, 44.

After the several flow lines have been constructed, transit time alongthem may be determined by application of the formula where t is transittime, K is a constant, X is distance along the dew line, and V ispotential at any point along the flow line.

In this practice of the invention ex is measured along the .fiow linebetween two successive points at which potential has been determined sothat the potential drop 6V between them is known. For example, if 6X is1 cm. and 6V is .05 volt, at or transit time between the two points isor 2 0K speotively to those located on the electrolytic model with theprobe. Another form of linkage is illustrated in Pig. 4. It comprises anelectrolytic model 45in the form of a basin of insulating materialshaped to correspond to the for this purpose those electrodes which havethe smallest potential drop in the series resistors connecting them tothe transformer.) The probe is connected to the input of a cycle bandpass amplifier 40, the'ground potential side of which is connected tothe potentiometer common point, i. e. the junction of thetwo decadeboxes.

The sum of the resistances of the two decade boxes is maintainedconstant at say The pool of electrolyte is system undergoinginvestigation, say a wet gas field, and containing a pool ofelectrolyte. Electrodes 46, ll, 48 corresponding to extraction wells andan electrode 50 corresponding to an injection well, project into thepool from the bottom and receive current from'a source 5|, connectedbetween them. The extraction electrodes areinparallel. 1

Avertical probe 52 is supported above the pool in contact therewith andis mounted'on one end of a horizontalfarm or longitudinal slider '53}The other end of thearm carries a vertical marker 54. 'Achart or-map 55corresponding to the model in shape and orientation is disposed belowthe marker head.

The arm 53 is slidable longitudinally in a holder or lateral slider 55which rests on a horizontal'supporting rail 51 running at right anglesto the arm-. The' holder slides along the rail, so that the probe may bemoved to any portion of the pool with the marker occupyingacorresponding position above the chart.

As in the case of the apparatus of Figs. 1 and 2, means is provided fordetermining the potential found by the probe at any point in the pool ofelectrolyte. Thus a potentiometer 58 consist ing oi two adjustableresistances 58A, 58B in series is connected between the injectionelectrode 50 and one of the extraction electrodes 46. The probe isconnected to the common point of the potentiometer through agalvanometer 59.

Adjustment of potentials at the extraction electrodes to simulate anydesired set of extraction rates at the corresponding wells may beaccomplished by means of variable resistances 60, GI, 62, connectedrespectively to the extraction electrodes 46, 41, 48.

The operation 01' the apparatus of Fig. 4 is the same as that of Figs. 1and 2. The sum of the resistances in the potentiometer is kept constant,but by adjusting the ratio of the two, the potential of their commonpoint is set at a desired percentage of the total potential drop acrossthe two resistances. With current flowing through the electrodes of themodel and with the resistances associated with the electrodes adjustedto simulate any selected set of field operating conditions, the pool ofthe model is explored with the probe to locate the equipotential linecorresponding to the percentage voltage drop at which the midpoint isset. When the probe occupies a point on this equipotential line thegalvanometer will read a minimum value.

To consider a specific application of the invention to the mapping ofconditions in a wet gas field, reference is made to Fig. 5. This is aplan view 01 the isopachous map or chart 55 of Fig. 4, with thepositions of the extraction wells indicated by points A, "A, A, and thepositions of the injection well indicated by the point "A. For a givenset of flow conditions, as established by the potential gradients setbetween the corresponding electrodes, a plurality of equipotential linesare plotted. Actually, the number of equipotential lines plotted isgreater than shown, the others having been eliminated in the interestsof simplicity. Flow lines are then constructed perpendicular to theequipotential lines. Transit times along the several flow lines are thencomputed as described above, and points of equal transit time areconnected together to establish so-called dry gas invasion fronts."These latter show the extent of invasion of the dry gas for the assumedset of conditions at the end of the particular transit time selected.

By readjusting the potential flow between injection and extractionelectrodes a number of times to simulate a corresponding set or assumedinjection and extraction rates for the actual wells, and plottinginvasion fronts as described above each time, the efiect of variousexploitation procedures may be determined in advance and that planchosen which will give optimum recovery. In short, the life history of awet gas field under any number of selected exploitation procedures maybe investigated in advance of exploitation to determine which one isbest.

I claim:

In apparatus for determining the condition of a system in which forceoperates at least approximately in accordance with Laplace's equationand having a three dimensional electrically conductive model analogousto the system to be investigated, with a conductive probe in contactwith the model and a marker disposed adjacent a chart spaced from themodel and a linkage connecting the marker to the probe for moving themarker on the chart to correspond to movement of the probe on the model,said linkage comprising an elongated arm carrying the marker and theprobe at its opposite ends, an elongated supporting rail disposedbetween the model and the chart at a right angle to the arm, and aslider slidably supporting the arm to permit longitudinal movement ofthe arm and slidably mounted on the rail to move longitudinally of therail.

ALEXANDER WOLF.

REFERENCES CITED The following references are of record in the file ofthis patent:

UNITED STATES PATENTS Number Name Date 1,825,855 Craig Oct. 6, 19311,919,215 Gunn July 25, 1933 2,368,217 Hayes Jan. 30, 1945 2,382,093Phelan Aug. 14, 1945 2,440,693 Lee May 4, 1948 OTHER REFERENCES ElectronOptics-Theoretical and Practicalby Myers. Published by D. Van NostrandCo. Pages 122 to 142 inclusive. (A copy is available in Division 54.)

Text Book Geophysical Exploration by Heiland 1940-chapter 10, pages681-706. (Copy of this text in Division 48.)

